I am following chapter 9 of the Rammer's book on Field Theory (which you can find here: https://www-thphys.physics.ox.ac.uk/talks/CMTjournalclub/sources/Rammer.pdf). I am referring to section 9.2.2, page 274, that deals with the derivation of the Dyson-Schwinger equation (of a cubic theory) using diagrammatic techniques.
I write the final result, Eq. (9.32): $$ \frac{\delta Z[J]}{\delta J_1}=G^{(0)}_{1\bar{1}}\left(\frac{1}{2} g_{\bar{1}23}\frac{\delta^2}{\delta J_3\delta J_2}+J_{\bar{1}}\right)Z[J]\,, $$ where $Z[J]$ is the generating functional, $J_i$ is the $i$th component of the source term, $G^{(0)}_{12}$ is the propagator of the free (Gaussian) theory and $g_{123}$ is the coupling of the considered cubic theory. The repeated indices Einstein convention is used. An important remark is that the factor $\frac{1}{2}$ pops out at page 264 as a combinatorial factor,
"the device to make the bare vertex diagram (here a 3-vertex) appear with no combinatorial factor".
In particular, the author says
"an $N$-line vertex carries an explicit prefactor $1/(N − 1)!$".
This is of course a convention.
What I don't understand is the connection with the Dyson-Schwinger equation I know: $$ 0=\left(\frac{\delta S[\delta/\delta J]}{\delta \varphi_1}-J_1\right)Z[J]\,, $$ where $Z[J]=\int D\varphi\exp{(-S[\varphi]+J\varphi)}$ and $$S[\varphi]=S_0[\varphi]+V[\varphi]=\frac{1}{2}(G^{(0)}_{12})^{-1}\varphi_1\varphi_2+V[\varphi].$$ I can make the connection between the two bringing the bare propagator from the RHS of the first equation on the LHS and recognising $\frac{\delta}{\delta\varphi_1}S_0[\delta/\delta J]Z[J]$. The result for the potential $V[\varphi]$ will be: $$ -\frac{\delta}{\delta\varphi_1}V[\delta/\delta J]=\frac{1}{2}g_{123}\frac{\delta^2}{\delta J_3\delta J_2}\qquad\Rightarrow\qquad V[\varphi]=-\frac{1}{3!}g_{123}\varphi_1\varphi_2\varphi_3\,, $$ which is in contrast with the convention $1/(N-1)!=1/2$ for the cubic theory. I think the problem is only a combinatorial factor. Does anyone have a suggestion?
